determine the quadratic equations whose graph is given the vertex is (1,1) and the y intercept is (0,-7).
1 answer:
The vertex form of a quadratic equation is
... y = a(x -h)² + k . . . . . . where the vertex is (h, k) and "a" is some multiplier
Then using the given information, we can find "a".
... -7 = a(0 -1)² +1 = a+1
... -8 = a
The quadratic equation is
... y = -8(x -1)² +1 . . . . vertex form
or
... y = -8x² +16x -7 . . . . standard form
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Check out the attached image below. Use polynomial long division (see figure 1) or synthetic division (figure 2) to find that the quotient is x-8
This tells us the slant asymptote is y = x-8
The presence of the slant asymptote means we do not have a horizontal asymptote. As x gets larger and larger, and heads to infinity, then y approaches the line y = x-8; rather than y approaching some single fixed value.
The vertical asymptote is x = -4 because this x value makes the denominator x+4 equal to 0. We cannot divide by zero.
Figure 3 shows the graph, which I have done using GeoGebra (a free graphing tool). This is also attached below.
